2. DECIMAL NUMBERS 2-1 Extending the place value system 2 2-2 Is it a decimal point? 5 2-3 The size of decimal numbers 6 2-4 Scales with decimal numbers 7 2-5 Decimal accuracy 8 2-6 Rounding decimal numbers 9 2-7 Fractions and decimals 12 2-8 Recurring decimals 13 2-9 Adding and subtracting decimals 14 2-10 Multiplying decimals 15 2-11 Dividing by a decimal 16 2008, McMaster & Mitchelmore 1
3 719 10 = 371 r 9 (where r is an abbreviation for remainder ) 9 The remainder is 9 out of 10. As a fraction, this is: 10 719 = 700 + 10 + = 7 + 1 + 10 100 Activity 2-1 Extending the place value system To say that our number system is a decimal system means that is a place value system with a base of 10. 3 719 x 10 = 37190 37190 = 30 000 + 7000 + 100 + 90 Multiplying by 10 results in each digit moving one place to the left because each term in the expansion is multiplied by 10. 3 719 x 100 = 371 900 371 900 = 300 000 + 70 000 + + 900 Multiplying by 100 results in each digit moving 2 places to the left because each term in the expansion is multiplied by 10 twice. 371 900 10 = 37 190 37 190 10 = 3 719 When you divide them by 10 each digit moves one place to the right. 3 719 100 = 37 r 19 The remainder is 19 out of 100. As a fraction, this is: 3 719 1 000 = 3 r 719 The remainder is 719 out of. As a fraction, this is: 9 9 19 100 719 2008, McMaster & Mitchelmore 2
Hundred thousands Ten thousands Thousands Hundreds Tens Ones Tenths Hundredths Thousandths 3 7 1 9 0 0 3 7 1 9 0 3 7 1 9 3 7 1 9 3 7 1 9 3 7 1 9 Yes. With this double line in the table (but without the column headings) you could tell that the numbers in the last four rows of the table are all different. 3 719 10 = 371.9 371.9 10 = 37.19 37.19 10 = 3.719 3.719 x 10 = 37.19 3.719 x 100 = 371.9 3.719 x = 3719 2008, McMaster & Mitchelmore 3
NUMERAL NUMBER 2 0 6 0 0. 0 0 0 0 0 20 thousand, 6 hundred 0 2 0 6 0. 0 0 0 0 0 2 thousand and sixty 0 0 2 0 6. 0 0 0 0 0 2 hundred and 6 0 0 0 2 0. 6 0 0 0 0 20 and 6 tenths 0 0 0 0 2. 0 6 0 0 0 2 and 6 hundredths 0 0 0 0 0. 2 0 6 0 0 206 thousandths 0 0 0 0 0. 0 2 0 6 0 206 ten thousandths 0 0 0 0 0. 0 0 2 0 6 206 hundred thousandths The zeros which must always be written down if a number is to stay the same size are: any zeros between non-zero digits and any zeros between a non-zero digit and the decimal point. Note: If a decimal is less than 1, people usually write a zero in the ones column (before the decimal point) because otherwise, the decimal point may not be noticed. This zero in the ones column however, does not change the value of the number. When the last word is hundred, the 6 is in the hundreds place. When the ending of the word is ty the 6 is in the tens place. When the last word is tenths, the 6 is in the tenths place. NUMBER NUMERAL 7 tens 70 7 tenths 0.7 7 hundredths 0.07 45 hundredths 0.45 4 thousand and 5 4005 4 thousand, 5 hundred 4500 45 hundreds 4500 68 thousandths 0.068 3 ten thousandths 0.0003 507 ten thousandths 0.0507 507 hundred thousandths 0.00507 2008, McMaster & Mitchelmore 4
Activity 2-2 Is it a decimal point? 1) Australians used to think that petrol selling for 109.5 cents per litre was expensive. Yes. It is a decimal point. Reason: The 5 means 5 tenths of a cent. 2) Tani set her alarm for 5.30 in the morning so she would not miss her flight home from Thailand. No. It is not a decimal point. Reason: The 3 does not mean 3 tenths of an hour. Neither does the 30 mean 30 hundredths of an hour. It means 30 minutes (which is 5 tenths of an hour). 3) In April 2001, the Australian dollar fell as low as 0.483 US dollars. Yes. It is a decimal point. Reason: The 0.483 means 483 thousandths of a dollar (which is 48.3 cents). 4) In Sydney in 2008, Eamon Sullivan set a new world swimming record of 21.56 seconds for 50 metres freestyle. Yes. It is a decimal point. Reason: The.56 means 56 hundredths of a second. 5) For homework, Laura was given exercise 4.9, exercise 4.10 and exercise 4.11 in her mathematics textbook. No. It is not a decimal point. Reason: The exercises are numbered consecutively. As decimal numbers, 4.9 is larger (not smaller) than 4.10 so they would be out of order. 2008, McMaster & Mitchelmore 5
The size of decimal numbers Activity 2-3 3.8 > 3.72 6.516 < 6.8325 0.4 > 0.26 8.924 > 8.63 0.536 > 0.4 2.83 < 2.84 3.08 < 3.7 0.47 > 0.316 7.0 = 7 0.65 < 0.7 4.62 < 4.736 0.0 = 0 5.92 > 5.4815 0.55 = 0.550 15.373 > 15.37 0.5 < 0.75 0.941 = 0.94100 2.75 > 2.74 0.7 > 0 0.6 > 0.0006 2008, McMaster & Mitchelmore 6
Scales with decimal numbers Activity 2-4 5 There are 10 major units are between 0 and 5. The temperature represented by one major unit is 0.5 C. 8 major units together represent 4 C. There are 20 minor units between 4 and 5. 2 minor units together represent 0.1 C. 4 The temperature represented by one minor unit is 0.05 C. 2.5 C 2.05 C 0.1.2.3.4.5.6.7.8.9 1 0.2.4.6.8 1 1.2 1.4 1.6 1.8 2 1.25 C 0.9 C 0.01.02.03.04.05.06.07.08.09.1 0.002.004 006.008.01 1 1.02 1.04 1.06 1.08 1.1 0 0.1 0.12 0.14 0.16 0.18 0.2 2008, McMaster & Mitchelmore 7
Activity 2-5 Decimal accuracy 0kg 200 400 600 800 1kg 200 400 600 800 2kg Precision: 40 g =.04 kg The precision of most rulers is 1 millimetre. If a line was 100.6 mm long, Tim would say it was 101 mm. If a line was 99.4 mm long, Tim would say it was 99 mm. The accuracy of the kitchen scales above is 20 g Accuracy is sometimes described by the words correct to. If a measurement is 400 ± 0.5 mm, it is correct to the nearest millimetre. If a measurement is 400 ± 5 mm, it is correct to the nearest 10 mm. A measurement of 0.4 m is correct to the nearest 0.1 m (which is 10 cm). A measurement of 0.40 m is correct to the nearest centimetre (0.01 m). A measurement of 0.400 m is correct to the nearest millimetre (0.001 m). 2008, McMaster & Mitchelmore 8
Rounding decimal numbers Activity 2-6 Dane decides to buy the pack of 12 and sell them to his friends. Cost per can in this pack (to the nearest cent) is $1.02. Dane has to pay $12.30 for the pack if he pays in cash. For Dane, cost per can (to the nearest cent) is $1.03. Dane s friends said they d buy a can from Dane for the price he paid. Each friend will pay Dane $1.05 in cash. If you were one of Dane s friends, it would cost the same ($1.05) for a can if you bought it from Dane or if you paid the shop for a can in cash because $1.07 is rounded down to $1.05. A number with 4 decimal places might be rounded: to 3 decimal places (i.e. to the nearest thousandth) or to 2 decimal places (i.e. to the nearest hundredth) or to 1 decimal place (i.e. to the nearest tenth). The shorter marks The point 0.9625 lies between the two short marks that represent the numbers 0.962 and 0.963. The longer marks on this scale represent hundredths. The point 0.9625 lies between two of these long marks. The numbers represented by these two long marks are 0.96 and 0.97. 0.9625 lies closer to 0.96. So 0.9625 becomes 0.96 to 2 d.p. The two long thick marks on this scale represent tenths. The numbers represented by these long thick marks are 0.9 and 1.0. 0.9625 lies closer to 1.0. So 0.9625 becomes 1.0 to 1 d.p. If the last of the required number of digits is kept as it is, the number has been rounded down. You decide whether to round up to the next number by looking at the last digit. If it is 5, round up. 2008, McMaster & Mitchelmore 9
Rounding 2.0954 to 1) 3 d.p. 2.095 2) 2 d.p. 2.10 3) 1 d.p. 2.1 4) 0 d.p. 2 No. Mr Skite s result was not correct to the nearest $10. He went wrong because he should have decided on the accuracy he would use (i.e. the number of places he would round to) from the beginning. If he rounded to the nearest $10, this would make the sheep $90 each. 1) Sue had necklaces to make for the school fair. She knew that it took her 40 minutes to make each one and that she had up to 7 hours in which she could make them. How many necklaces could she make? Answer: 420 40 = 10.5. This is rounded to 10 necklaces. Reason: Half a necklace cannot be sold so it shouldn t be counted. 2) 21 people in a youth group want to go lawn bowling together. They need drivers to take them to the lawn bowling club. If each driver can legally take 4 passengers in their car, how many drivers are needed? Answer: 21 4 = 5.25. This is rounded to 6 drivers. Reason: Another driver is needed to drive the one remaining person. 3) Steve and his friends earned $278.50 from busking together. Steve banked the money into his account. Then he paid the others their share in cash. How much money will Steve have left? Answer: $278.50 4 = $69.625. This is rounded to $69.65. $278.50-3x $69.65 = $69.55 Reason: Cash is rounded up to the nearest 5 cents. 2008, McMaster & Mitchelmore 10
A statistician predicts that Australia s population in 2050 will be 29.549085 million. An example of a headline is: 30 million Aussies by 2050. (Newspapers round numbers to make them sound more sensational as well as making them easy to read.) Tim carefully cut a string 16.036 m long into 8 equal pieces. Tim would say the length of each piece was 2.005 m (to show he was being accurate to the nearest mm). Kyron and Shirong went to watch a 100m cycling race. They decided to calculate the speed of the winning cyclist. With his watch, Kyron measured a time of 5 seconds. So he calculated the winner s speed to be 100 5 = 20 m/s. With his watch, Shirong measured a time 6 seconds. On his calculator he made the division 100 6 = 16.66666667 m/s. Shirong said his answer was more accurate because it had more decimal places in it. No. Shirong was not right because a calculation cannot produce a number that is more accurate than the measurements used in the calculation. Shirong should have rounded his answer to 17 m/s. Some examples of contexts for the division 100 6 which have the following answers: 1) 16 The number of 6 L containers you could fill with 100 L. 2) 17 The number of half cartons needed to hold 100 eggs. 3) 16.65 The cash received by each of 6 people when $100 is divided between them. 2008, McMaster & Mitchelmore 11
Fractions and decimals 1) 0.3 3 2) 0.03 3 3) 0.030 10 100 59 509 4) 0.059 5) 0.509 6) 5.9 7) 5.09 509 8) 5.009 5009 9) 0.5090 100 714 Activity 2-7 = 0.714 714 = 0.714 623 59 8 1) 0.623 2) 0.059 3) 0.08 100 1 1 4) 0.5 5) 0.25 6) 3 0.75 2 4 4 1 3 5 7) 8 0.125 8) 8 0.375 9) 8 0.625 If your calculator shows 10 digits (and doesn t round automatically) your answers will be: 1) 1 0.333333333 2) 2 0.666666666 3 3 3) 2 0.153846153 4) 7 0.162790697 13 43 30 Multiplication of the answer by the denominator does not quite give you the value of the numerator because the calculator can only show a limited number of decimal places on its screen. Converting 2 to a decimal: 3 The digit 2 is being carried each time a division is made. 59 10 5090 0 If you divide by a number and then multiply by the same number, your result is the number you started with (eg. 5 8 x 8 = 5) because multiplication and division are opposite operations. This division process can be repeated infinitely. It all right to put a decimal point after the 2 and write zeros after it because this does not change its value. 2008, McMaster & Mitchelmore 12
Activity 2-8 Recurring decimals 0. 8 7 5 To write 7 as a decimal: 8) 7. 0 6 0 4 0 8 As a decimal: 7 = 0.875 8 You know that the decimal you calculated above is an exact value because there was no final remainder. As a decimal correct to 2 d.p.: 7 = 0.88 8 To write 6 as a decimal: 7 0. 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 7) 6. 0 4 0 5 0 1 0 3 0 2 0 6 0 4 0 5 0 1 0 3 0 2 0 6 0 4 0 5 0 6 = 0.85714285714286 (to 14 d.p.) 7 No. Could you never write out a decimal number that is exactly 6 because the numbers carried would keep repeating in a cycle. 7 The digits written in decimal places before they started repeating themselves were 857142. 6 = 0.8571428571428571425714257 (to 25 d.p.) 7 Written as a recurring decimal: 6 = 0.857142 7 If a fraction is written as a recurring decimal, you do not need to write that it is correct to a certain number of d.p. because you can write a recurring decimal to any number of places you like. Unordered decimals 0.82 0.802 0.082 0.82 0.82 0.08 0.082 0.082 Ordered decimals 0.082 0.082 0.082 0.08 0.802 0.82 0.82 0.82 2008, McMaster & Mitchelmore 13
Activity 2-9 Adding and subtracting decimals When decimals with different numbers of decimal places are to be added or subtracted, their digits should not be right-aligned because this puts digits with different values in the same column. Questions INCORRECT ALIGNMENT CORRECT ALIGNMENT 20.35 20.35 29.7 29.7 0.008 0.008 7 256 7 256 7 306.058 1) Alice, Brianna and Chloe each start knitting a scarf. After a week they measure the lengths of their knitting. Alice says hers is 98 cm, Brianna says hers is 975 mm and Chloe says hers is 1 m and 5 mm. The total length of their knitting is 2.960 m. 0.98 0.975 1.005 2.960 2) Andrew s ten-pin bowling ball weighs 6.8 kg. While on holiday, he doesn t have his own ball but there are others he can play with. An orange red ball weighs 6.398 kg and a blue ball weighs 7.190 kg. The weight closest to 6.8 kg is 7.190 kg. 6.800 7.190-6.398-6.800 0.402 0.390 3) In the middle of the night, a mother accidentally gave her screaming baby 0.0125 L of paracetamol instead of 0.00125 L. The extra volume of paracetamol given to the baby was 0.01125 L. 0.01250-0.00125 0.01125 2008, McMaster & Mitchelmore 14
Activity 2-10 Multiplying decimals 8 x 70 = 560 8 x 7 x 10= 560 90 x 20 = 1800 9 x 10 x 2 x 10 = 1800 1300 x 20 = 26000 13 x 10 x 10 x 2 x 10 = 26000 For these calculations, the rule Take off the zeros, multiply, then add the zeros works because by taking off a zero you are dividing the number by 10. By adding a zero you are multiplying the number by 10. The multiplication compensates for the division. 4 x 9 10 = 3.6 4 x 0.9 = 3.6 9 10 x 3 10 = 0.27 0.9 x 0.3 = 0.27 423 x 2 10 10 10 10 = 0.0846 4.23 x 0.02 = 0.0846 0.2 x 0. 03 = 0.006 0.20 x 0.03 = 0.006 0.20 x 0.030 =0.006 The 3 calculations above should all have the same value because the two numbers being multiplied have the same value. A rule that will help you multiply decimals correctly: Multiply each of the numbers by a number of 10s until the resulting numbers can be written without a decimal point. Count the number of times you needed to multiply both numbers by 10. Do the multiplication algorithm with the resulting numbers. Divide the answer to your algorithm by the number of 10s you previously multiplied by. 1) 657 x 3.4 2) 0.860 x 50.9 3) 0.019 x 9.8450 = 2233.8 = 43.7740 = 0.1870550 = 43.774 = 0.187055 6 5 7 8 6 0 9 8 4 5 0 x 3 4 x 5 0 9 x 1 9 2 6 2 8 7 7 4 0 8 8 6 0 5 0 1 9 7 1 0 4 3 0 0 0 0 9 8 4 5 0 0 2 2 3 3 8 4 3 7 7 4 0 1 8 7 0 5 5 0 2008, McMaster & Mitchelmore 15
Activity 2-11 Dividing by a decimal 56 7 = 8 560 70 = 8 180 2 = 90 1800 20 = 90 2600 13 = 200 260 000 1300 = 200 The two calculations in the same row give the same answer because when the dividend and the divisor are multiplied by the same number, these operations compensate each other. 40 8 = 5 4 0.8 = 5 6480 2 = 3240 648 0.2 = 3240 132 12 = 11 1.32 0.12 = 11 The first column of calculations was easier. 1) 67.893 0.6 678.93 6 2) 7.080 0.05 708 5 3) 9.8 0.34 980 34 4) 9.3 0.068 9300 68 1) 323 3.4 2) 1.76 0.32 3) 1.633 0.23 = 95 = 5.5 = 7.1 9 5 5.5 7.1 34) 3 2 3 0 32)1 7 6.0 23) 1 6 3.3 3 0 6 1 6 0 1 6 1 1 7 0 1 6 0 2 3 1 7 0 1 6 0 2 3 0 0 0 2008, McMaster & Mitchelmore 16